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	\begin{document}
	
	\begin{table}[htbp] 
  \centering
  \caption{\bf  Estimating  Number of Risk Factors ($r=1$)}
\medskip
 \begin{tabular}{llllllllll}
   & $N$ & $T$ & $PC_{p1}$ & $IC_{p1}$ & $AIC_1$ & $BIC_1$ & $PC (0.10)$  & $PC (0.25)$ & $PC (0.45)$ \\ \hline 
    & 5     & 2     & 2     & 2     & 2     & 2     & 1       & 1        & 1        \\
 & 15    & 2     & 2     & 2     & 2     & 2     & 1       & 1        & 1        \\
 & 30    & 2     & 2     & 2     & 2     & 2     & 1       & 1        & 1        \\
 & 100   & 2     & 2     & 2     & 2     & 2     & 1       & 1        & 1        \\
 & 200   & 2     & 2     & 2     & 2     & 2     & 1       & 1        & 1        \\
 & 500   & 2     & 2     & 2     & 2     & 2     & 1       & 1        & 1        \\
 & 1,000  & 2     & 2     & 2     & 2     & 2     & 1       & 1        & 1        \\
 & 3,000  & 2     & 2     & 2     & 2     & 2     & 1       & 1        & 1   \\
 & 5     & 3     & 3     & 3     & 3     & 3     & 1.28    & 1.132    & 1.052    \\
 & 15    & 3     & 3     & 3     & 3     & 3     & 1.316   & 1.08     & 1        \\
 & 30    & 3     & 3     & 3     & 3     & 3     & 1.256   & 1.024    & 1        \\
 & 100   & 3     & 3     & 3     & 3     & 3     & 1.248   & 1.008    & 1        \\
 & 200   & 3     & 3     & 3     & 3     & 3     & 1.196   & 1        & 1        \\
 & 500   & 3     & 3     & 3     & 3     & 3     & 1.184   & 1        & 1        \\
 & 1,000  & 3     & 3     & 3     & 3     & 3     & 1.136   & 1        & 1        \\
 & 3,000  & 3     & 3     & 3     & 3     & 3     & 1.072   & 1        & 1      \\  \hline     
\end{tabular}
    \parbox{5.25in}{\footnotesize{This table presents the estimated number of factors, $ \tilde{r}$ (average across Monte Carlo iterations).  The true number of factors is $r=1$ and I search for
 $ 0 \le k \le T =2,3 $. The $ e_{it}$ are assumed iid $N(0,1)$ across $i$ and $t$. Columns 3 to 6 correspond to the competing criteria $IP_{p1}, IC_{p1}, AIC_1, BIC_1$ (see  Bai and Ng (2002)[Section 5] for details). Columns 7  to 9 correspond to my large-$N$ criterion $ PC(\epsilon_2) =  ({T \over T-k}) V(k) +k  g(N) $  with 
\[
g(N)  =  \epsilon_3  {  (log(N))^{\epsilon_1} N^{\epsilon_2} \over T \sqrt{N}  },
\]
with $ \epsilon_1=0 $, $  \epsilon_2  =0.1, 0.25, 0.45$, $ \epsilon_3 =1$.}}
\label{Table9MCOA}
\end{table}



\begin{table}[htbp] 
  \centering
  \caption{\bf  Estimating  Number of Risk Factors ($r=1$, cont.)}
\medskip
\begin{tabular}{llllllllll}
   & $N$ & $T$ & $PC_{p1}$ & $IC_{p1}$ & $AIC_1$ & $BIC_1$ & $PC (0.10)$  & $PC (0.25)$ & $PC (0.45)$ \\ \hline 
 & 10    & 5     & 5.16  & 5.11  & 5.13  & 5.19  & 1.11    & 1.01     & 1        \\
  & 30    & 5     & 5.03  & 5.03  & 5.03  & 5.04  & 1.06    & 1        & 1        \\
  & 100   & 5     & 5.05  & 5.03  & 5.03  & 5.05  & 6       & 1        & 1        \\
  & 200   & 5     & 5.07  & 5.06  & 5.06  & 5.07  & 6       & 1        & 1        \\
  & 500   & 5     & 5.07  & 5.07  & 5.07  & 5.07  & 6       & 1.2      & 1        \\
  & 1,000  & 5     & 5.05  & 5.05  & 5.05  & 5.05  & 6       & 6        & 1        \\
  & 3,000  & 5     & 5.01  & 5.01  & 5.01  & 5.01  & 6       & 6        & 1        \\
  & 10    & 10    & 8     & 8     & 8     & 8     & 1.02    & 1        & 1        \\
  & 30    & 10    & 8     & 8     & 8     & 8     & 1       & 1        & 1        \\
 & 100   & 10    & 8     & 8     & 8     & 8     & 1       & 1        & 1        \\
 & 200   & 10    & 8     & 8     & 8     & 8     & 1       & 1        & 1        \\
 & 500   & 10    & 8     & 8     & 8     & 8     & 1       & 1        & 1        \\
& 1,000  & 10    & 8     & 7.51  & 8     & 8     & 1       & 1        & 1        \\
& 3,000  & 10    & 8     & 1     & 8     & 8     & 1       & 1        & 1        \\
 & 10    & 30    & 8     & 8     & 8     & 8     & 1       & 1        & 1        \\
 & 30    & 30    & 6.15  & 1     & 8     & 8     & 1       & 1        & 1        \\
 & 100   & 30    & 1.5   & 1     & 8     & 3.6   & 1       & 1        & 1        \\
 & 200   & 30    & 1     & 1     & 8     & 1     & 1       & 1        & 1        \\
 & 500   & 30    & 1     & 1     & 4.84  & 1     & 1       & 1        & 1        \\
 & 1,000  & 30    & 1     & 1     & 1.29  & 1     & 1       & 1        & 1        \\
 & 3,000  & 30    & 1     & 1     & 1     & 1     & 1       & 1        & 1        \\
 & 10    & 60    & 8     & 8     & 8     & 8     & 1       & 1        & 1        \\
 & 30    & 60    & 3.54  & 1     & 8     & 8     & 1       & 1        & 1        \\
 & 100   & 60    & 1     & 1     & 8     & 2.18  & 1       & 1        & 1        \\
 & 200   & 60    & 1     & 1     & 8     & 1     & 1       & 1        & 1        \\
 & 500   & 60    & 1     & 1     & 4.78  & 1     & 1       & 1        & 1        \\
 & 1,000  & 60    & 1     & 1     & 1     & 1     & 1       & 1        & 1        \\
 & 3,000  & 60    & 1     & 1     & 1     & 1     & 1       & 1        & 1        \\
 & 10    & 120   & 8     & 8     & 8     & 8     & 1       & 1        & 1        \\
 & 30    & 120   & 1.13  & 1     & 8     & 8     & 1       & 1        & 1        \\
 & 100   & 120   & 1     & 1     & 8     & 4.2   & 1       & 1        & 1        \\
 & 200   & 120   & 1     & 1     & 8     & 1     & 1       & 1        & 1        \\
 & 500   & 120   & 1     & 1     & 8     & 1     & 1       & 1        & 1        \\
 & 1,000  & 120   & 1     & 1     & 1.43  & 1     & 1       & 1        & 1        \\
 & 3,000  & 120   & 1     & 1     & 1     & 1     & 1       & 1        & 1    \\ 
  \hline
\end{tabular}
    \parbox{5.25in}{\footnotesize{This table presents the estimated number of factors, $ \tilde{r}$ (average across Monte Carlo iterations).  The true number of factors is $r=1$ and I search for
 $ 0 \le k \le T $. The $ e_{it}$ are assumed iid $N(0,1)$ across $i$ and $t$. Columns 3 to 6 correspond to the competing criteria $IP_{p1}, IC_{p1}, AIC_1, BIC_1$ (see  Bai and Ng (2002)[Section 5] for details). Columns 7  to 9 correspond to my large-$N$ criterion $ PC(\epsilon_2) =  ({T \over T-k}) V(k) +k  g(N) $  with 
\[
g(N)  =   \epsilon_3  {  (log(N))^{\epsilon_1} N^{\epsilon_2} \over T \sqrt{N}  },
\]
with $ \epsilon_1=0 $, $  \epsilon_2  =0.1, 0.25, 0.45$, $  \epsilon_3  = 1 $.}}
\label{Table10MCOA}
\end{table}




 \begin{table}[htbp] 
  \centering
  \caption{\bf  Estimating  Number of Risk Factors ($r=3$)}
\medskip
\begin{tabular}{llllllllll}
   & $N$ & $T$ & $PC_{p1}$ & $IC_{p1}$ & $AIC_1$ & $BIC_1$ & $PC (0.10)$  & $PC (0.25)$ & $PC (0.45)$ \\ \hline 
& 10    & 5     & 5.05  & 5.05  & 5.05  & 5.05  & 4.38    & 3.57     & 2.23     \\
& 30    & 5     & 5.01  & 5     & 5     & 5.01  & 6       & 5.98     & 2.43     \\
& 100   & 5     & 5.01  & 5     & 5     & 5.01  & 6       & 6        & 3.6      \\
& 200   & 5     & 5.05  & 5.05  & 5.05  & 5.05  & 6       & 6        & 4.22     \\
& 500   & 5     & 5.05  & 5.02  & 5.01  & 5.05  & 6       & 6        & 5.11     \\
& 1,000  & 5     & 5.03  & 5.02  & 5     & 5.03  & 6       & 6        & 5.7      \\
& 3,000  & 5     & 5.08  & 5.07  & 5.06  & 5.08  & 6       & 6        & 5.85     \\
  & 10    & 10    & 8     & 8     & 8     & 8     & 4.4     & 3.39     & 2.04     \\
  & 30    & 10    & 8     & 8     & 8     & 8     & 6.19    & 3.23     & 1.8      \\
  & 100   & 10    & 8     & 8     & 8     & 8     & 6.76    & 2.87     & 1.76     \\
  & 200   & 10    & 8     & 8     & 8     & 8     & 6.5     & 2.9      & 1.87     \\
  & 500   & 10    & 8     & 8     & 8     & 8     & 5.77    & 2.91     & 1.86     \\
  & 1,000  & 10    & 8     & 8     & 8     & 8     & 5.29    & 2.95     & 1.95     \\
  & 3,000  & 10    & 8     & 8     & 8     & 8     & 4.39    & 2.99     & 1.94     \\
  & 10    & 30    & 8     & 8     & 8     & 8     & 4.22    & 2.88     & 1.88     \\
  & 30    & 30    & 6.34  & 2.93  & 8     & 8     & 3.61    & 2.91     & 1.94     \\
  & 100   & 30    & 3.09  & 3     & 8     & 4.46  & 3.04    & 3        & 2.07     \\
  & 200   & 30    & 3     & 3     & 8     & 3     & 3       & 3        & 2.15     \\
  & 500   & 30    & 3     & 3     & 5.52  & 3     & 3       & 3        & 2.26     \\
  & 1,000  & 30    & 3     & 3     & 3.03  & 3     & 3       & 3        & 2.38     \\
  & 3,000  & 30    & 3     & 3     & 3     & 3     & 3       & 3        & 2.55     \\
  & 10    & 60    & 8     & 8     & 8     & 8     & 4.15    & 2.79     & 1.95     \\
  & 30    & 60    & 4.18  & 2.98  & 8     & 8     & 3.02    & 2.95     & 2.03     \\
  & 100   & 60    & 3     & 3     & 8     & 3.41  & 3       & 3        & 2.23     \\
  & 200   & 60    & 3     & 3     & 8     & 3     & 3       & 3        & 2.32     \\
  & 500   & 60    & 3     & 3     & 5.49  & 3     & 3       & 3        & 2.62     \\
  & 1,000  & 60    & 3     & 3     & 3     & 3     & 3       & 3        & 2.7      \\
  & 3,000  & 60    & 3     & 3     & 3     & 3     & 3       & 3        & 2.76     \\
  & 10    & 120   & 8     & 8     & 8     & 8     & 3.53    & 2.78     & 1.98     \\
  & 30    & 120   & 3.01  & 2.99  & 8     & 8     & 3       & 2.97     & 2.07     \\
  & 100   & 120   & 3     & 3     & 8     & 4.91  & 3       & 3        & 2.37     \\
  & 200   & 120   & 3     & 3     & 8     & 3     & 3       & 3        & 2.45     \\
  & 500   & 120   & 3     & 3     & 8     & 3     & 3       & 3        & 2.81     \\
  & 1,000  & 120   & 3     & 3     & 3.12  & 3     & 3       & 3        & 2.9      \\
  & 3,000  & 120   & 3     & 3     & 3     & 3     & 3       & 3        & 2.96  \\  \hline
\end{tabular}
    \parbox{5.25in}{\footnotesize{This table presents the estimated number of factors, $ \tilde{r}$ (average across Monte Carlo iterations).  The true number of factors is $r=3$ and I search for
 $ 0 \le k \le T $. The $ e_{it}$ are assumed iid $N(0,1)$ across $i$ and $t$. Columns 3 to 6 correspond to the competing criteria $IP_{p1}, IC_{p1}, AIC_1, BIC_1$ (see  Bai and Ng (2002)[Section 5] for details). Columns 7  to 9 correspond to my large-$N$ criterion $ PC(\epsilon_2) =  ({T \over T-k}) V(k) +k  g(N) $  with 
\[
g(N)  =  \epsilon_3  {  (log(N))^{\epsilon_1} N^{\epsilon_2} \over \sqrt{N}  },
\]
with $ \epsilon_1=0 $, $  \epsilon_2  =0.1, 0.25, 0.45$, $  \epsilon_3  = 1 $.}}
\label{Table11MCOA}
\end{table}


    \begin{table}[htbp] 
  \centering
  \caption{\bf  Estimating  Risk Factors}
\medskip
 \begin{tabular}{lllll}
 &$ N=10$ &  $N=100$ & $N=1,000$ & $N=3,000$  \\ \hline
$T=2$  & - & - & -  & -  \\
$T=5$ &  0.942    &  0.996   & 0.999  &   0.999 \\
$T=50$  & 0.955 &    0.996 &    0.999 &      0.999 \\ 
\hline
\end{tabular}
    \parbox{4.0in}{\footnotesize{This table presents the sample correlations between $\tilde{\bf F}_t $ and ${\bf F }_t$ for $ 1 \le t \le T $ (average across Monte Carlo iterations). }}
\label{Table12MCOA}
\end{table}




    \begin{table}[htbp] 
  \centering
  \caption{\bf  Estimating  Risk Exposures}
\medskip
 \begin{tabular}{lllll}
  & $N=10$ & $ N=100 $& $N=1,000$ & $N=3,000$  \\ \hline
$T=2$  &   0.610 &   0.667 &    0.674 &      0.675 \\
$T=5$ &  0.875 &   0.905 &   0.907 &   0.907 \\
$T=50$ &  0.989 &    0.991 &    0.992 &     0.992  \\
\hline
\end{tabular} 
  \parbox{4.0in}{\footnotesize{This table presents the  sample correlations between $\tilde{\boldsymbol{ \lambda}}_i $ and $\boldsymbol{ \lambda }_i$ for $ 1 \le i \le N $ (average across Monte Carlo iterations).}}
%\label{Table13MCOA}
\end{table}


	
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\end{document}
